Paper 4. A Seventeen-Card Witness

Osman, Mohamed A.

Paper 4. A Seventeen-Card Witness

Seventeen cards on 27 symbols; 136 pairs meet and none of 80,730 five-sets covers

Mohamed A. Osman · ORCID 0009-0004-5912-999X · independent researcher · 2026-10-19

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Abstract. An explicit pairwise intersecting 6-uniform family of seventeen cards on 27 symbols with transversal number six. All 136 card pairs intersect, and exhaustive checking of all 80,730 five-symbol subsets finds no transversal. Hence g(6) is at most 17. The check needs nothing but the standard library and uses no result from the other three papers.

No priority is claimed for any result, and this is not presented as a settled answer. The lower bound is computer-assisted and has not been independently reviewed by a human or by a proof assistant. AI tools were used throughout the work and its review; every computational claim is regenerated by a script or certified by a proof file in code/ and results/ of the repository. Licence: CC BY 4.0 (text), MIT (code).